Some random thoughts on game theory, behavioural economics, and human behaviour

Friday, 28 July 2017

Risk aversion or loss aversion

Suppose you offer someone called Albert a gamble - if the toss of a coin comes up heads then you pay him £100 and if it comes up tails he pays you £100. The evidence suggests that most people will not take on that gamble. If Albert also turns down the gamble, what does that tell you about Albert's preferences?

One thing we can conclude is that Albert is risk averse. In particular, the gamble was fair because Albert's expected payoff was 0 and, by definition, if someone turns down a fair gamble then they are exhibiting risk aversion. It is hard to argue with a definition and so we can conclude that Albert is risk averse. The more interesting question is why he displays risk aversion?

The micro-economic textbook would tell us that it is because of diminishing marginal utility from money. A diagram helps explain the logic. Suppose that Albert has the utility function for money depicted below. In this specific case I have set the utility of £m as the square root of m. Notice that the utility function is concave in the sense that it gets flatter for larger amounts of money. This is diminishing marginal utility of money - the more money Albert has the less he values more.

Suppose that Albert has £500. If he does not take the gamble then his utility is 22.36. If he takes the gamble then he can end up with either £400 or £600. The former gives him utility 20 and the latter 24.49. The expected utility is midway between this, i.e. 22.25. Crucially the expected utility of the gamble, 22.25, is less than not taking on the gamble, 22.36, and so Albert does not gamble. As the bottom figure shows we get this result because the utility function is concave. That means the utility of not gambling - on the blue line - lies above the expected utility of gambling - on the red line.

There is though a problem with this standard story, formally demonstrated by Matthew Rabin in a 2000, Econometrica paper 'Risk aversion and expected utility theory: A calibration theory'. If Albert is risk averse over a relatively small sum of money like 100 with an initial wealth of £500 then he would have to be unbelievable risk averse over large gambles. Basically, he would never leave his front door. If diminishing marginal utility of money is not the explanation for Albert's risk aversion then what is?

The most likely culprit is loss aversion. Now we have to evaluate outcomes relative to a reference point rather than a utility function over wealth. It seems natural to think that Albert's reference point is £500. That would mean winning the gamble is a gain of £100 and losing the gamble is a loss of £100. Crucially, the evidence suggests that people typically interpret a loss as worse than a gain is good. This is shown in the next figure.
In this case everything is judged relative to the status quo of £500. Having more than £500 is a gain and less than £500 is a loss. The steeper value function below £500 captures loss aversion. The crucial thing to observe is that loss aversion effortlessly gives concavity of the value function around the status quo. So, Albert would prefer to not gamble and have value 0 than to gamble and have either +5 or -10 with an expected value of -2.5. Loss aversion has no problem explaining risk aversion over small gambles.

So, what can we conclude from all this? The first thing to tie down is the definition of risk aversion. Standard textbooks will tell you that risk aversion is turning down a fair gamble. That to me seems like a fine definition. So far, so good. Confusion (including in academic circles) can then come from interpreting what that tells us. Generations of economists have been educated to think that risk aversion means diminishing marginal utility of money. It need not. We have seen that loss aversion can also cause risk aversion. And there are other things, like weighting of probabilities that can also cause risk aversion. It is important, therefore, to consider different possible causes of risk aversion.
And it is also important to recognize that there is unlikely to be some unified explanation for all risk aversion. We know that people do have diminishing marginal utility of money over big sums of money. We know people are loss averse over surprising small amounts of money. We also know that people are bad at dealing with probabilities. All of these factors should be put in the mix. So, Albert might buy house insurance because of diminishing marginal utility of money, not try the new cafe for lunch because he is loss averse, and buy a lottery ticket because he overweights small probabilities.